Math Problem Statement

1. Tentukan himpunan penyelesaian dari 3x - 5 < |x + 3a + 2b|. 2. Tentukan himpunan penyelesaian dari |2a + b + (2a+3b)/(2x+a+b)| > a + 2b.

Solution

The image contains two mathematical problems where aa and bb represent the last two digits of a student identification number (NIM). In the example given, for a NIM ending in "61", a=6a = 6 and b=1b = 1.

The two questions are:

  1. Find the solution set of 3x5<x+3a+2b3x - 5 < |x + 3a + 2b|.

  2. Find the solution set of 2a+b+2a+3b2x+a+b>a+2b\left| 2a + b + \frac{2a + 3b}{2x + a + b} \right| > a + 2b.

Now, let's break down and solve these questions using the example values of a=6a = 6 and b=1b = 1.

Question 1:

Solve 3x5<x+3a+2b3x - 5 < |x + 3a + 2b|.

With a=6a = 6 and b=1b = 1, the equation becomes: 3x5<x+3(6)+2(1)=x+18+2=x+203x - 5 < |x + 3(6) + 2(1)| = |x + 18 + 2| = |x + 20| We can break this inequality into two cases based on the absolute value:

  • Case 1: x+200x + 20 \geq 0, so 3x5<x+203x - 5 < x + 20 3xx<20+53x - x < 20 + 5 2x<25x<252=12.52x < 25 \quad \Rightarrow \quad x < \frac{25}{2} = 12.5

  • Case 2: x+20<0x + 20 < 0, so 3x5<(x+20)3x - 5 < -(x + 20) 3x5<x203x - 5 < -x - 20 3x+x<20+53x + x < -20 + 5 4x<15x<154=3.754x < -15 \quad \Rightarrow \quad x < -\frac{15}{4} = -3.75

Therefore, the solution to the inequality is x<3.75x < -3.75 or x<12.5x < 12.5, but x+200x + 20 \geq 0 only holds when x20x \geq -20.

Thus, the final solution set is: x(20,12.5)x \in (-20, 12.5)

Question 2:

Solve 2a+b+2a+3b2x+a+b>a+2b\left| 2a + b + \frac{2a + 3b}{2x + a + b} \right| > a + 2b.

Substitute a=6a = 6 and b=1b = 1 into the inequality: 2(6)+1+2(6)+3(1)2x+6+1>6+2(1)\left| 2(6) + 1 + \frac{2(6) + 3(1)}{2x + 6 + 1} \right| > 6 + 2(1) Simplify: 12+1+12+32x+7>6+2=8\left| 12 + 1 + \frac{12 + 3}{2x + 7} \right| > 6 + 2 = 8 13+152x+7>8\left| 13 + \frac{15}{2x + 7} \right| > 8

Now, we need to solve the inequality: 13+152x+7>8\left| 13 + \frac{15}{2x + 7} \right| > 8 This also splits into two cases:

  • Case 1: 13+152x+7>813 + \frac{15}{2x + 7} > 8 138>152x+713 - 8 > -\frac{15}{2x + 7}

Do you want to explore further solutions or have clarifications?


Here are 5 follow-up questions for practice:

  1. How does the absolute value affect the range of solutions?
  2. Can you derive a general solution for inequalities involving absolute values?
  3. What are the properties of rational inequalities?
  4. How does adding constants inside absolute values impact the solution set?
  5. What strategies help simplify complex fractions in inequalities?

Tip: Always check the domain of your variable in inequalities before concluding the solution set.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Absolute Value
Rational Expressions

Formulas

3x - 5 < |x + 3a + 2b|
|2a + b + (2a + 3b)/(2x + a + b)| > a + 2b

Theorems

Absolute Value Theorem
Properties of Inequalities

Suitable Grade Level

Grade 10-12